An infintesimal Lorentz transform and its inverse inverse (primed to unprimed) can be written as, x g x x g x ;
in which: O( ) 0 O( )
[I.1]
The objective of the problem is to find the generators of the Lorentz transformation in a simplified way (compared with Chapter 11.7 of Jackson). The simplification is achieved by considering finite Lorentz transformations as a sequence of infinitesimal ones. The generators of the l atter are obtained in the following. (a) show, from the definition of the inverse, that . Inverting both sides of the definition of the inverse: Taking the inverse of both sides of x ( g ) x
from [I.1], and using the definition of the inverse provided, we use the obvious obv ious Minkowski-metric inverse g g , and write, x ( g
) x ( g ) x
[I.2]
Focus our attention on the second term x , which, by definition of the inverse, becomes,
( x ) ( g [ x ]) g [( g ) x ] g ( g )g x
Putting [I.3] back into [I.2], and realizing that by definition of inverses
x
x
[I.3]
, we have,
x g (g )g x g g (g )g x x x g
[I.4]
This means we identify as the two bracketed terms at the end of [I.4]. writing this statement, and doing
algebra, but algebra, but carefully illustrating the steps we are taking above the “=” signs so that we can warm up to tensor algebra, g
( g
) g ( g
) g
g
g
g
g
g
b) show from the preservation preservation of the norm that
( g 0
) g (g
g (
g
)
) g
[I.5]
.
Use the Lorentz transform [I.1], and just use the same pedestrian manipulations that were throughout [I.5], and you get, x x x x ( g ) x x ( g ) x g x ( g ) x g ( g ) x
( g ) g ( g ) x g x ( g )( g ) g x x
( g )( g g g ) x x g g x x x x g g x x [I.6] x x g g x x x x g g x x x x g x x x x x x
g x x x x x x x x x x
c) by writing the transformation in terms of contravariant components on both sides of the equation, show that
is
equivalent to the matrix L S K . Using the exponential representation A e 1 L O( L ) , and the and the L
infintesimal lorentz transform [I.1], we get two separate definitions for x ; equating them, we get,
2
x g
x g g x g x ;
x A x
L x L g [I.7]
This last result of [I.7] then leads to,
L g
g g L g gLg L L is an antisymmetric matrix
[I.8]
That L is antisymmetric means we have six linearly independent basis “vectors” that the antisymmetric matrix is a linear combination of. The matrix-structure shown on Jackson p. 546 follows. Jackson says that L S K is an abbreviation for this structure; QED.
